Abstract

As the island of our knowledge grows, so does the shore of our ignorance.

John Wheeler

© Dr M D Magee
Contents Updated: Thursday, 28 November 2002
Monday, 12 March 2007

Problems with the Hot Big Bang

Conventional ideas fail to explain or even describe the ultimate origin of the universe. The universe today has regions that could never in their history have been in causal contact in the Big Bang model. These regions are moving away from one another so fast that any information, even travelling at the speed of light, could not cover the distance between them. This “horizon problem” makes it difficult to account for the uniformity of the cosmic background radiation.

Disastrously, the Big Bang model does not explain the origin of large-scale structures like galaxies. Edward R Harrison and Yakov B Zel’dovich showed that large-scale structures could appear from small fluctuations in the density of matter in an otherwise homogeneous early universe, but no one could say where these fluctuations came from. They just had to be assumed.

Then there is the “flatness problem”. The curvature of the universe depends on its density, its weight in relation to its volume, yet the spatial geometry of the observable universe is flat. Flatness is unstable. A slight variation will cause a bend and the universe will then become curved, either positively like a ball or negatively like a saddle, bearing in mind that these are two dimensional analogies of what is happening in three spatial dimensions. The Big Bang theory predicts the universe will get more curved as time passes, but since the universe seems to be flat, it must have started out flat—to within one part in 10⁶⁰, essentially free of imperfections. Many cosmologists think this unlikely, but the evidence is that the density of the universe is close to the critical value needed for the cosmos to be flat. The hot Big Bang model depended too much on initial conditions.

Some process early in its life must have smoothed out the imperfections expected in a Big Bang to make the universe flat. In 1980, Alan H Guth proposed that a release of energy happened in the first minute or so of the universe’s existence. He called it “inflation”. Guth suggested inflation to solve those problems of cosmologcal theory. It would have lasted for a fleeting second, but in it the universe would have increased in size by 10⁶⁰ times before it slowed down to its present rate of expansion. The “inflationary universe” model resembled the hot Big Bang except that the universe had that brief but rapid period of expansion.

Inflation solves the horizon problem because the observed universe emerges from a region small enough to permit causal contact. The flatness problem vanishes because its huge expansion smooths out irregularities across the universe, flattening it. Inflation stretches out any wrinkles in the curvature of the universe, just as inflating a balloon smooths out its surface. The density-fluctuation problem is also solved—the scenario predicts that the sudden expansion would have locked in quantum fluctuations that could have seeded the formation of large-scale structures.

But why would it happen? In the hot Big Bang model, the matter content of the universe is a uniformly distributed plasma or dust. Guth’s model considers a cold universe of matter consisting of scalar-field particles, which are not the stuff of everyday life, but they do arise naturally in many theories, and they are believed to be the main form of matter in the high energy conditions of the early universe. They make gravity become a repulsive force, and inflation occurs. The decay of the scalar-fleld matter producing the expansion heated the cold universe to a very high temperature, by the end of the inflationary moment. Subsequent evolution follows the path of the hot Big Bang model—the universe expanded and cooled, and the residual heat is detectable as the cosmic background radiation.

This provides an explanation for the origin of the density fluctuations that would have seeded galaxies. All types of matter are subject to such quantum effects, but for most purposes the fluctuations are so small as to be totally insignificant. The rapid expansion of the universe during inflation magnified these initially insignificant microscopic fluctuations, transforming them into macroscopic changes in density. The much slower expansion in the hot Big Bang model is incapable of producing this effect.

Inflation does not relieve the observed state of the universe of all dependence on assumptions about initial conditions. It would have occurred only if the scalar field began with a large, approximately constant energy density, equivalent, at least for a brief time, to Einstein’s cosmological constant. So, inflation rests on assumptions about initial conditions.

Testing Inflation by Weighing the Universe

Remarkably, inflation left space filled with matter at precisely the critical density for flatness. Could this be confirmed by observation? It could not. Astronomers had been seeking ways of weighing the universe for decades, but could not get consistent results. The measured density was less than the critical value, though teasingly close to it. Then cosmologist, Jeff Peterson (New Scientist 2000), and others successfully weighed the universe, and their result confirmed scientific hypotheses of how the universe began. They used the faint afterglow of the Big Bang—the cosmic microwave background (CMB)—which can still be detected in every part of the sky.

The microwave background radiation has in it the relics of the time just before the universe became visible—the very time that the background radiation records. A mere 100,000 years after the big bang, a plasma of electrons and hydrogen and helium ions filled all space making it opaque because the electrons blocked the paths of photons in the plasma. Fluctuations in the density of the universe were heightened by the consequent pull of gravity of the more dense regions, tending to form galaxies. But, the pressure of the light, the photons impacting the electrons of these denser regions had the opposite effect of spreading them. The net effect was not an immediate equilibrium but an initial vibration of the plasma which eventually would have damped out into an equilibrium.

The universe continued vibrating like this until its temperature, after 300,000 years from the big bang, fell to 4,500 degrees. Temperature measures heat energy—the energy of motion that particles have—and at this temperature, the electrons did not have enough energy to resist the electrical attraction of the heavier hydrogen and helium ions. Atoms formed, removing many of the free electrons and thus allowing more space for photons to move in without hitting an electron. Light could pass through it, and so the universe became transparent. It is the light that was released at this moment that is the CMB, light that has been travelling for 13 billion years, and it contains a photograph of the last vibrations of the cosmos before it became transparent, displayed as bright and dim regions of the sky. This pattern yields the density of the universe.

Smaller regions of space vibrated faster, and larger regions more slowly. The largest regions had not completed a vibration when the universe became transparent, but the smallest patches had vibrated through several cycles. Different regions were at different stages in these cycles of compression under gravity and spreading under the pressure of light, but the later cycles were being damped out. Those that were at their maximum compression in their first cycle when the universe became transparent should be the clearest in the sky, and theory shows how big these regions should have been 300,000 years after the big bang. Allowing for the expansion of the universe and its density yields the state of these regions in the sky today. A higher density universe, acting like a lens, makes these regions of compression look bigger to us than a low density universe would because matter curves the path of the light by its gravity. How much bigger they look compared with how big they were expected to be is a measure of the weight of the universe.

During the 1990s, observations showed signs of these expected vibrations, but the resolution of the instruments was not good enough for any accuracy. In 1998, Peterson using a telescope at the South Pole, and the Mobile Anisotropy Telescope in the Atacama Desert achieved the needed resolution, but only in a narrow field of vision. Even so, the half-cycles seemed to be present. Then in 2000, results from two balloon-borne telescopes, Boomerang and MAXIMA, reported. Boomerang mapped out one per cent of the sky, enough for accurate interpretation, and MAXIMA mapped out another quarter per cent.

Researchers use mathematical techniques to filter out the frequencies of the vibrating regions, yielding a “power spectrum”, a distribution of the amount of each frequency present in the sample of sky measured. The results from Boomerang and MAXIMA have a clear first peak in the power spectrum, corresponding to regions that had gone through half a cycle to their maximum compression when the universe became transparent. From the frequency of this peak, researchers can work out how big the half-cycle regions now look to us on the sky. The position of the peak shows how much material there is in the universe, and hence how much it weighs. More peaks in the specrum give even greater accuracy, and NASA has the Microwave Anisotropy Probe to do it by mapping the entire sky. ESA plans to launch the Planck satellite in 2007. Planck will image the entire sky with a sensitivity better than MAP, and with twice the resolution. The measured density of the universe is within about 6 per cent of the critical value. The universe is indeed flat, and the theorists and their ideas about inflation seem to be right.

Problems remain. Results from optical observers and the microwave background telescopes are different, though both may be right. Cosmic background measurements include the effects of energy as well as matter, whereas the optical instruments show visible matter like stars and galaxies. Energy exerts a gravitational pull on the paths of CMB photons just as matter does, the missing component of the universe’s weight might be dark energy, though nobody is sure even what it could be. Nor is enough visible matter observed, suggesting there is something called dark matter.

Even though these results seem to confirm inflation, how it occurred or at what temperature are not known. It might be that the laws of gravity are wrong. Stacy McGaugh of the University of Maryland has shown that dark matter disappears if gravity is slightly higher at low accelerations than Newton or Einstein predict. Even then dark energy is needed to explain the cosmic observations.

How Did the Universe Begin?

What happened before inflation? How did the universe actually begin?

The precursor state of the Big Bang is some unstable vacuum state in the infinite past. It is not convincing. Besides the pre-Big Bang idea there are two other hypotheses of boundary conditions of the universe—the tunnelling hypothesis, and the no boundary proposal. Following the expansion of the universe backward in time to the pre-inflation era, the size of the universe tends to zero, and the strength of the gravitational field and the energy density of matter tend to infinity. The universe was a singularity, a region of infinite curvature and energy density at which the known laws of physics break down.

In the 1960s, Steven Hawking and Roger Penrose of the University of Cambridge proved singularities are a consequence of any model of the expanding universe extrapolated backward in time, under current assumptions. But these assumptions are not valid. There are faults in the classical theory of general relativity close to the beginnings of time. A singularity does not physically occur. The volume of a singularity shrinks to small dimensions, and, near it, space-time becomes highly curved causing classical general relativity to break down. It has to be superseded by a better theory. What is this theory? Under such circumstances, one must appeal to the theory of small things—quantum theory.

So, extrapolation backward in time eventually takes the universe down to a size at which quantum theory has to be applied to it. Attempts to do this are called quantum cosmology. If the universe was initially small in size, then quantum effects must then have been important on a cosmic scale. In the 1960s, Bryce S DeWitt, Charles W Misner and John A Wheeler showed how quantum mechanics might be applied to the entire universe. Their work was not taken seriously until the 1980s, when classical theories of cosmology began to fail.

Niels Bohr, Erwin Schrödinger, Werner K Heisenberg, Paul A M Dirac, and others, in the early twentieth century, developed quantum mechanics. Motion is not deterministic as in classical mechanics but probabilistic. Position and momentum do not generally have definite values in quantum mechanics. The wave function encodes the probabilistic information about position, momentum and energy, and is found by solving the Schrödinger equation.

For a single-point particle, the wave function is an oscillatmg field spread throughout physical space. At each point in space, the function has an amplitude and a wavelength. The square of the amplitude is proportional to the probability of finding the particle at that position. The wavelength is related to the momentum of the particle, but an uncertainty in the measured position and momentum will always exist. As the measurement of position, gets more precise, the value of the momentum gets more indefinite. This is Heisenberg’s uncertainty principle, and is a consequence of the wave nature of particles.

In quantum mechanics, a system can never have an energy of exactly zero. The kinetic and potential energy cannot both be exactly zero because the particle would then be motionless and so its position and momentum would both be known and that is forbidden. It has a ground state in which the energy is as low as can be but not zero (ZPE, zero point energy). The particle must be wiggling around enough to preserve the uncertainty principle, so it cannot be still. Such fluctuations also prevent the orbiting electron from crashing into the nucleus. The electrons have an orbit of minimum energy from which they cannot fall into the nucleus without violating the uncertainty principle. Remember, galaxies form from “ground-state fluctuations”, in the inflationary universe.

Quantum Cosmology

Like quantum mechanics, quantum cosmology attempts to describe a system fundamentally in terms of its wave function. One can find the wave function of the universe by solving an equation called the Wheeler-DeWitt equation, which is the cosmological analogue of the Schrödinger equation. In the simplest cases, the spatial size of the universe is the analogue of position, and the rate of the universe’s expansion represents the momentum.

The main problem of quantum cosmology is the lack of a quantum theory of gravity. All attempts to quantize Einstein’s general relativity have failed. The theory of gravity, general relativity, says that at the singularity, space becomes infinitely small and the energy density infinitely great. To look beyond such a moment requires a quantum theory of gravity. However, proponents of a theory of “superstrings” claim it to be a consistent, unified quantum theory of all four forces of nature and thus is, or at least contains, a quantum description of gravity. Even more contentious is that quantum mechanics applies to the entire universe at all times and to everything in it. This is the fundamental assertion of quantum cosmology.

In the Copenhagen convention developed by N Bohr, the observer was considered to be separate from the particle being observed. Only the wave function of the particle was considered, the observer being too big to have its wave function affected, and the observation changed the wave function, or “collapsed” it in the jargon. Hugh Everett III of Princeton said that, for cosmology, one universal wave function described both macroscopic observers and microscopic systems, with no fundamental division between them. A measurement is an interaction between one part of the universe and another, and the wave function predicts the change observed. The wave function does not collapse, but evolves smoothly described by the Schrödinger equation for the entire system.

Then, Everett made a remarkable discovery. Observation caused the universe to “split” into enough copies of itself to cover all possible outcomes of the measurement. Much discussion has occurred about whether these multiple universes are real in some sense or are just possible outcomes. Murray Gell-Mann and Hartle discount any probabilities that are not close to 0 or 1, that is are impossible or certain, on the grounds that, while an electron can have a small chance of being present anywhere, a universe must either be present or absent.

From this, at certain points in space and time—typically, but not always, when the universe is large—the wave function of the universe behaves classically, classical space-time is predicted, and the wave function gives probabilities for the set of possible classical behaviors of the universe.

Other regions, such as those close to classical singularities, exist in which no such prediction is possible. Paul Davies assures us that S Augustine of Hippo, in the fifth century, proclaimed the world was made “not in time, but simultaneously with time”. Time began with the cosmic origin. Modern science clarifies S Augustine’s speculation, based on what we now know about the nature of space, time, and gravitation. Einstein taught that time and space are altered by gravity. In the extreme conditions of the early universe, the distortion of space and time seemed to have been infinite, and formed an impassable boundary. Time did not stretch back for all eternity. Near a singularity, space and time do not exist. There is a “quantum fuzz”, still describable by known laws of quantum physics but not by classical laws. In quantum cosmology, classical initial conditions cannot be imposed on a region in which classical physics is not valid, such as near the initial singularity. If the Big Bang was the beginning of time itself, any discussion of what happened before it, or what caused it is meaningless.

Some physical events do not have a well-defined cause. An abrupt and uncaused event can occur within the scope of scientific law, at the small scale of things that mean quantum laws have to be used. Nature can be spontaneous. The decay of a radioactive nucleus happens once and for all at one moment not another, and it is impossible to know when. All anyone can do is give the probability of decay in a certain time. This is the half-life. This uncertainty is a basic part of quantum reality—of Nature.

One of many wave functions possible (the many solutions to the Wheeler-DeWitt equation) has to be singled out. Hartle and Hawking, Linde, and Vilenkin have made proposals intended to pick out a unique Wheeler-DeWitt wave function for the universe.

Hawking and Hartle’s Explanation of Universal Origins

James B Hartle, Stephen W Hawking, Andrei D Linde and Alexander Vilenin formulated laws of the conditions that existed at the Big Bang. Combined with the laws that govern the evolution of the universe, these initial conditions could explain all cosmological observations and would resolve the problems of conventional cosmology. Quantum cosmology is an advance over hot Big Bang cosmology in defining the initial conditions of the universe as a ground state (a state of minimum excitation) whose fluctuations lead to inflationary expansion and galaxy formation. Halliwell and Hawking (1985) state:

The boundary conditions imply that these modes start off in the ground state… Thus the proposal that the quantum state of the Universe is defined by a path integral over compact four-metrics seems to be able to account for the origin of structure in the Universe: it arises, not from arbitrary initial conditions, but from ground-state fluctuations that have to be present by the Heisenberg uncertainty principle.

In 1984, Stephen W Hawking and James Hartle had realized that uncertainty applies to space-time when quantum effects are considered. Time and space become indistinguishable in the quantum fuzz, but space can progressively change into time in a short but continuous process, causing the start of the universe. They generalized Richard P Feynman’s path integral, or sum-over-histories method, in which summing over histories is mathematically equivalent to solving the Schrödinger equation, to quantum cosmology. By describing the universe as a wave function, “the wave function of the universe”, they gave a probabilistic and noncausal explanation of why the universe exists.

Calculating the wave function of the universe by summing over some class of histories for the universe is equivalent to solving the Wheeler-DeWitt equation. The precise solution obtained depends on how the class of histories summed over is chosen. Hartle and Hawking chose ones which smoothly contracted to zero. Their idea is called the no-boundary proposal because the sum is over geometries that have no boundary except for the final end, which corresponds to the present universe. In this, Hawking says he is using the Anthropic Principle, in that only a universe like ours is a feasible outcome, so that many alternative paths can be discarded. Hawking says:

The anthropic answer is that two spatial dimensions are not enough for complicated structures, like intelligent beings. On the other hand, four, or more, spatial dimensions would mean that gravitational and electric forces would fall off faster than the inverse square law. In this situation, planets would not have stable orbits around their star, nor would electrons have stable orbits around the nucleus of an atom. Thus intelligent life, at least as we know it, could exist only in four dimensions. I very much doubt we will find a non-anthropic explanation… It seems that the Anthropic Principle really requires the no boundary proposal, and vice versa.

There are many other solutions but they do not give a world suitable for people. The smooth closing off takes place in “imaginary” time—time expressed as a complex number involving the square root of minus one, in which it resembles a spatial dimension—but the process occurs nevertheless, so imaginary time has a real enough effect. Convert back into real time only, and boundaries and singularities appear. There is no Big Bang but an incredibly rapid but progressive conversion of space into time out of the quantum fuzz. When is time sufficiently different from the other dimensions to be recognizably time? It is impossible to say, so there is no clear first moment of time. Universal time within the quantum fuzz extends back for eternity, yet observable time began when it distinguished itself sufficiently from the quantum fuzz! L Z Fang and Z C Wu, (Quantum Cosmology), conclude from this work no “First Cause” is needed in cosmology because…

…in principle, one can predict everything in the universe solely from physical laws.

Hawking adds:

If the no boundary proposal is correct, He (God) had no freedom at all to choose initial conditions.

The origin of the universe from nothing need not be unlawful, unnatural or unscientific. It need not have been a supernatural event. This hypothesis has been criticized, especially by those with an interest in defending God the Maker, but Quentin Smith a philosopher of science at Westerm Michigan University has countered them.

The wave function of the universe takes account of all possible influences and offers unqualified conditions for occurrences of states of the universe and of the universe as a whole. It is about the whole universe, and so nothing outside the universe can influence it, otherwise it would not be a wave function of the universe but a wave function of only part of the universe.

The wave function of the minutely small initial universe, like that of any tiny object, is given by psi², where psi is the wave function of a three-dimensional space (because it is at a prescribed time) which is the first state of the new universe at the earliest possible time, 10^-43 seconds, the Planck time—at the moment of the Big Bang. From the probability of this temporal slice of space, the other states of the universe can be calculated. The tiny space is so tightly curved it is the shape of a smooth sphere, and the constituents of it are so tightly packed they are uniform in distribution within it. The probability amplitude is summed over all the possible amplitudes that can fulfil these conditions—all possible histories of a finite universe—and so yields an overall probability of any such an object forming, a probability conditional only on the mathematical properties of the wave functions of possible universes. Physical evidence—the background radiation shown by COBE, the homogeneity and isotropy of the universe on the large scale, an inflationary period, and the critical density being near to one—confirms that Hartle and Hawking’s wave function corresponds with the real universe we observe.

So, the probability that the universe exists depends on nothing but natural mathematical properties and the boundary conditions of the wave function. Inasmuch as wave functions give an accurate account of the behaviour of natural objects at the quantum level, the hypothesis explains the beginning of the universe based on natural laws only. Hawking says in A Brief History of Time, the wave function gives in principle the probabilities of the histories of intelligent organisms:

Each history in the sum over histories will describe not only the space-time but everything in it as well, including any complicated organisms like human beings who can observe the history of the universe.

Christians seek a gap for God, as they inevitably do, claiming that God sets the boundary conditions, or declare the method to be nothing more than a mathematical trick. You could say the same about a great deal of mathematics that turned out to be describing something real. Sharon Begley wrote:

The world follows rules, rules that are fundamentally mathematical, rules that humans can figure out. Humans invent abstract mathematics, basically making it up out of their imaginations, yet math magically turns out to describe the world. Greek mathematicians divided the circumference of a circle by its diameter, for example, and got the number pi, 3.14159… Pi turns up in equations that describe subatomic particles, light and other quantities that have no obvious connections to circles.

Newsweek July 20, 1998

But waves are related to the circle because they are caused by cycles. Quantum mechanics itself is a prime example. It is also called wave mechanics. Besides Pi, imaginary numbers appear throughout physics and no Christians ever complained about them until they dispensed with God. Imaginary numbers and Pi help describe wave motion, and waves are real enough. For all these fixated Christians know, imaginary time might be the spiritual dimension that they have been talking about for centuries. If it turned out to be that, they would quickly begin approving of Hawking. They might eventually canonize him!

Believers think the existence of the universe was at the whim of God, but the formulation of Hartle and Hawking does not need any supernatural states or entities to be involved. Believers think the probability of a universe existing independently of God is zero, but, because our universe exists, it has a probability of being caused by God of unity. The quantum formulation does not give such idealistic likelihoods. The probabilities are fractional, and so cannot be compatible with ideas of God. Even if God has ordained the physics behind the quantum mechanical formulation, He is leaving the beginning of the universe to chance, and the ideas of believers accordingly are false.

From the beginning on there is nothing deterministic about the state of the universe. All future configurations of the entities within the initial point space exist with equal probabilities, according to Hawking in an early paper (1976). In our case, we can see what came out, but there is no particular propensity for the initial point to explode into an eventual universe that contains intelligent life. It is actually unlikely. If God caused the universe to contain intelligent life, the initial state must have favoured the existence of intelligent life.

Believers find themselves doing what they have always done in desperation—trying to patch up their beliefs piecemeal. The incessant need for Christians to find excuses for this and that on God’s behalf ought to be sufficient to undermine belief, but that it often is not shows how remarkably irrational believers are. The belief in an almighty God is a belief that God could have made the universe in any way He fancied, and He fancied making it begin in a single point which “exploded” to form what we now see. Curiously, He chose the very way that any evolutionary universe would have had to begin—from the simplest possible beginning, as Smith calls it. The evidence of a Big Bang implies that the universe began from something a minimal step away from nothing. Why should God want to fool everyone into thinking the universe could have started without cause out of nothing, for nothing must be the only state that could precede the simplest of all beginnings, and nothing literally precludes the existence of something even if it is called God? No doubt He is testing His followers’ gullibility—something they call faith!

Some of the histories that have to be summed—those of intelligent organisms—include God communicating with human beings or causing miracles, or the fancy that there is a God that does these things. To be universal, a wave function of the universe must include these supernatural phenomena, if they really happen. If a wave function of the universe does not refer to real or fancied divine activity, it cannot be universal, and cannot describe all possible histories of intelligent organisms.

The conundrum for believers is that they take God to be transcendent—He makes the universe from the outside—so believers must take the wave function of the universe to be wrong. For, them, quantum cosmology and their belief cannot both be true. They can adopt the scientific view, thereby rejecting their belief, or continue to believe thereby rejecting the sciences of quantum mechanics and cosmology. While quantum mechanics applied to the small scale of the nature we experience directly seems unchallengeable from its success, quantum cosmology is less sure and can legitimately be challenged still, but, hitherto, when religion has challenged science, it has been science that has been shown to be right.

Quantum Tunnelling

Uncertainty also leads to “tunnelling”. In classical mechanics, a ball at rest in a bowl will never be able to get out. In quantum mechanics, position is not sharply defined but is spread over a range, giving some probability, however small, that the ball will be found outside the bowl. Any such a particle (the ball) is said to have “tunnelled” through the energy barrier (the side of the bowl).

Tunnelling does not occur in real time. Mathematically, the particle penetrates the barrier in “imaginary” time. Because imaginary time is characteristic of tunnelling processes in quantum theory, another solution of the Wheeler-Dewitt equation is that the universe has tunnelled from nothing. Inflation and the Big Bang occurred after the tunnelling. The no-boundary wave function, however, does not have the general features normally associated with tunnelling. It gives high probability of a classical universe appearing with a large size and a low energy density. An ordinary tunnelling process would suppress a transition from zero to large size and give highest probability for tunnelling to small size that has a high energy density.

Linde and Vilenkin independently put forward a “tunnelling” proposal, a mathematical scheme to pick out a solution to the Wheeler-DeWitt equation with tunnelling properties, which allows the universe to tunnel into existence from nothing.

Both the no-boundary and tunnelling wave function indicate that space-time behaves classically when the universe is a few thousand times larger than the size at which the four forces of nature would be unified (about 10^-33 cm), in agreement with observation. When the universe is smaller, the wave function says classical space-time does not exist.

Time and Imaginary Time

Moreover, the universe might be unbounded and therefore never began and never ends. Stephen Hawking says:

So long as the universe had a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having no boundary or edge, it would have neither beginning nor end: it would simply be. What place, then, for a creator?

Time might not have been the same as ours when the universe was young and highly compressed. Suppose that the time of the universe is really:

where T is a constant, perhaps Planck’s time, and t is the time we now experience. T is very small (10^-43 sec), so the time we now experience is the same as universal time. When t = T, t_u is 0, and universal times shorter than T are negative. If T is the Planck time, then observed time does not start until then. When t is 0, the time we extrapolate to be that of the Big Bang, the universal time that these cosmological events are experiencing is really ( -∞ )! In universal time, the universe never has a beginning, because there is an eternity before “time” (the Planck time?) arrives. So, we are experiencing an illusion like the railroad tracks seeming to come together in the distance. When we walk along them they never do. If we could walk along universal time, the universe would never start, it would just always be there.

What is interesting about this equation is that it can be derived directly from the Theory of Relativity’s equation for time dilation. For bodies moving at speeds approaching the speed of light, time dilates. If v is the velocity of the moving object and c the velocity of light, then:

where t_v is an interval of time for the object moving at speed v, and t_o is the interval of time passed by the stationary observer observing the moving object. The actual measure of time for the moving object dilates so its time goes slower relative to the stationary observer. If time had dilated by a factor of 2, then each unit of time is twice that of the stationary observer, and in one unit of time to the traveller, the stationary observer will have passed two units of time.

If the observer were the moving object, the equation would remain the same because the motion is a relative motion between the two objects, it being arbitrary which one is assumed to be stationary. Anyway, the equation shows that time stops when some object is travelling at the speed of light because t_v =  ∞ . If the reciprocal of the term on the left hand side were the measure, it is not the factor by which the moving object has its time dilated, but the time that passes in terms of the stationary observer’s time:

The ratio, v/c is itself the ratio of two times, the times it takes the object to travel a specified distance because v = s / t, where s is an interval in space or, in other words, a prescribed distance. The relationship between velocity and this time interval is inverted so we can write the equation:

Sorting the terms gives this simple equation:

Using the difference of two squares, it reduces to the same form as the original hypothesised equation. When we trace back our observed time to zero at the Big Bang, universal time has the value t_c, the time that corresponds with T in our original surmise, the time we took to be Planck’s time, a fundamental unit of time. For universal time to reach zero, our observed time has to become imaginary. It suggests that there never was a Big Bang. On the miniscule scale, time is eternal. Perhaps that will actually please Christians [†]Christians like the idea of a point of creation, but they also like the idea that God lives forever. Since there is a time before time zero, an eternity of time elapsing in the virtual world before the world was created at time t_c, this is God’s time for the Christian. Then God created the world at time t_c, not at time zero! Of course, the virtual world requires no God, but Christians will find him in there all right. In truth, the idea of a virtual time before time began does tie in with the ideas of the Zoroastrians who had an eternal time and a historical time, and these can be perhaps vague intimations of t_v and t_o, here..

Why should it be? It seems that not only is space quantized, but so is time. That is why there is a maximum speed, the speed of light. It is the minimum time that any object can take to cross the smallest unit of space. The reason there is a minimum is that some oscillator is necessary for time to be measured, and no oscillator has the energy to oscillate fast enough to measure any shorter time. That is because energy is quantized. Planck showed that high energy quanta could not be filled, and so the amount of energy distributed among high energy oscilators actually falls away, explaining the distribution of black body radiation, it unexpectedly falling away at high frequencies (short times!). Infinite energy would be needed to measure zero time, and plainly that is impossible. It is the same reason and explanation as the infinite energy needed to equal the speed of light.

This links the large and the small scale theories, the Relativity and Quantum theories.

Added Note 12 Feb 2010

Marcus Chown wrote in the New Scientist that for seven years, German scientists have been looking for gravitational waves—ripples in space time caused by super dense astronomical objects such as neutron stars and black holes. GEO600 has not detected any gravitational waves so far, but might have made the most important discovery in physics for half a century.

For many months, the GEO600 team members had been scratching their heads over inexplicable noise that is plaguing their giant detector. Then, Craig Hogan, a physicist at the Fermilab particle physics lab in Batavia, Illinois, predicted the noise they were detecting. GEO600 has stumbled upon the fundamental limit of space time—the point where space time stops behaving like a smooth continuum and breaks into quanta. “GEO600 is being buffeted by the microscopic quantum convulsions of space time”, says Hogan. At this magnification, the fabric of space time becomes grainy and is ultimately made of tiny units rather like pixels, but a hundred billion billion times smaller than a proton. This distance is known as the Planck length, a mere 10^-35 metres.

A black hole’s entropy, synonymous with its information content, is proportional to the surface area of its event horizon, the theoretical surface that is the point of no return for infalling matter or light. Microscopic quantum ripples at the event horizon can encode the information inside the black hole, so there is no mysterious information loss as the black hole evaporates, as once was thought. The 3D information about a precursor star can be completely encoded in the 2D horizon of the subsequent black hole—not unlike the 3D image of an object being encoded in a 2D hologram.

Now the universe as a whole has a horizon too—the boundary from beyond which light has not had time to reach us in the 13.7 billion year lifespan of the universe. If space time is a hologram, then you can think of the universe as a sphere whose outer surface is papered in Planck length sized squares, each containing one bit of information. Its data is encoded in quanta of the dimensions of Plancks’s length at its surface.

The holographic principle says that the amount of information papering the outside must match the number of bits contained inside the volume of the universe. Since the volume of the spherical universe is much bigger than its outer surface, how could this be true? The Planck length is far beyond the reach of any conceivable experiment, so nobody dared dream that the graininess of space time might be discernable. Hogan realised that to have the same number of bits inside the universe as on the boundary, the world inside must be made up of grains bigger than the Planck length. Contrary to all expectations, it brings its microscopic quantum structure within reach of current experiments. So while the Planck length is too small for experiments to detect, the holographic "projection" of that graininess could be much, much larger, at around 10^-16 metres.

Hogan realised that the Anglo-German GEO600 experiment ought to be the most sensitive to what he had in mind. He predicted that if the experiment’s beam splitter is buffeted by the quantum convulsions of space time, this will show up in its measurements (Physical Review D, vol 77, p 104031).

This random jitter would cause noise in the laser light signal. Incredibly, I discovered that the experiment was picking up unexpected noise.

C Hogan.

What would it mean it if holographic noise has been found?

We would have directly observed the quantum of time. It’s the smallest possible interval of time—the Planck length divided by the speed of light.

So How Did the Universe Actually Begin?

Near singularities, the tunnelling and no-boundary wave functions tell us classical general relativity is not valid, and notions of space and time implied by the question are inappropriate. A universe with nonzero size and finite energy density appears from a quantum fuzz. After quantum creation, the wave function gives the probabilities of different evolutionary paths, one of which includes the inflation postulated by Guth. Both the no-boundary and tunnelling proposals seem to predict the conditions for inflation, eliminating assumptions about the scalar-fleld matter to drive the rapid expansion.

The inflation model assumes the inhomogeneous parts started out in their quantum mechanical ground state—the lowest possible energy state consistent with the uncertainty principle. But in 1985, Hawking and Jonathan J Halliwell (whose explanation of much of this appeared in the Scientific American of December 1991) showed this is a consequence of the no-boundary proposal. Suitable inhomogeneities emerge naturally from the theory. Everything must be smooth and regular on the bottom cap of the space-time tube. This condition implies that inhomogeneous fluctuations must be zero there. Evolving up the tube in imaginary time, the fluctuations grow and enter the real-time region as small as they could possibly be—as the quantum mechanical ground-state fluctuations demanded by the inflation model. The tunnelling proposal makes the same prediction, for similar reasons.

So, according to quantum cosmology, the universe appeared from a quantum fuzz, tunnelling into existence and thereafter evolving classically. The most compelling aspect of this picture is that the assumptions necessary for the inflationary universe scenario may be compressed into a single, simple boundary condition for the wave function of the universe.

How can one verify a law of initial conditions? An indirect test is to compare the predictions of the quantum models with the initial conditions needed for standard classical cosmological models. In this endeavor, quantum cosmologists can claim a reasonable degree of success. More direct, observational tests are difficult. Much has happened in the universe since its birth, and each stage of evolution has to be modelled separately. It is difficult to distinguish between effects that result from a particular set of initial conditions and those that derive from the evolution of the universe or from the modelling of a particular stage.

What is needed is an observation of some effect that was produced at the beginning of the universe but was insensitive to the subsequent evolution. In 1987, Leonid Grishehuk of the Sternberg Astronomy Institute in Moscow argued that gravitational waves may be the sought-after effect. Quantum creation scenarios produce gravitational waves of a calculable form and magnitude. Gravitational waves interact weakly with matter as they propagate through space-time. Therefore, when we observe them in the present universe, their spectrum may still contain the signature of quantum creation. Detecting gravitational waves is, unfortunately, almost impossible, and current attempts have failed.

As verifying quantum cosmology is difficult, no one can say whether the no-boundary or the tunnelling proposals are the correct ones for the wave function of the universe. An answer to the question, “Where did all this come from?” is still some way away, but, through quantum cosmology, cosmologists have formulated and addressed the question in a meaningful and interesting way.

A Theory of Everything

The Theory of Everything is the hypothesis that the four forces of Nature—the strong, weak, electromagnetic forces and gravity—are varieties of a single force that existed at the moment of the Big Bang. It is a hypothetical theory! No one has yet formulated it. The problem is uniting gravity with the other three, which can be united. When someone manages to unite all four, then physicists say we have a Theory of Everything!

The best candidate for developing a TOE is string theory. It tries to reconcile the two great theories of the twentieth century because they seem largely incompatible. Quantum mechanics said the world was statistical on a small scale, that behaviour depended on probability. General relativity described how space is warped by gravity on a large scale, and force is the response of particles moving in warped space. But these two concepts are not compatible. To answer some big questions in physics, such as what actually happened in the Big Bang, do you use the equations of general relativity because the scale is that of the size of the universe? Or do you use quantum mechanics because it is all began in such a small space? Solving the problem is the “theory of everything” (TOE), and has been the aim of cosmologists for the decades since Einstein. String theory has become the leading contender to fulfil Einsteins dream.

String theory was thought of in the late 1960s to describe the strong nuclear interaction, the force that stops protons flying apart in the nucleus of atoms. Protons are made of more fundamental particles called quarks held together by particles, called gluons, that transmit the strong nuclear force. Quarks and gluons were never found free, even when atoms were smashed apart in particle accelerators. String theory was intended to answer this. Gluons and quarks can be thought of as being the ends of a string of energy, and no string has just one end. Eventually quantum chromodynamics, a precise quantum mechanical description of the interaction between quarks and gluons was found more useful for the time being. Later, string theory was salvaged to explain not just the strong force but all the fundamental forces. Everything in the universe and all the forces acting upon them is made of tiny vibrating strings of energy.

In string theory, particles are not envisaged as little golf balls but little vibrating strings, 10^-33 meters long. A single string could seem to be any one of several particles depending on its state of vibration, particularly its vibrational frequency. The number of sustainable frequencies depends on how many half wavelengths can be fitted along the length of the string, so the range of possible frequencies is not continuous but goes up according to the whole number of half wavelengths that fit into the length of the string. The energy of the particle which manifests itself, then depends upon the frequency, and the energy goes up in quantum numbers corresponding to the half-wavelengths. One mode of vibration of a string yields an electron. Another yields a neutrino, and so on for the range of sub-atomic particles. The same is true of photons, bosons, gluons and would have to be true too of gravitons, these being the four binding particles of the four forces, exchange of which constitutes the force.

The trouble with string theory is that it gives too many particles, “far more than anyone wanted”, says Edward Witten of the IAS at Princeton. The possibilities exceed the number of particles in the universe, and the reason is that string theory, as presently conceived, needs 10 dimensions to work in, not the three, or four if time is included, that we observe. All of the dimensions not observed, are somehow hidden, or folded away. It is the large number of ways of selecting three from ten, and then the ways of folding up the remaining seven that gives the vast number of possible solutions to the equations of string theory. Any of this vast number could be a universe with different forces and constants from the one we live in, and different numbers of them.

Other solutions have other than three observable dimensions, and the numbers and types of bosons, photons and so on could be different. They might be lighter or heavier, or not even exist in some possible universes. Many of the solutions are unstable, so that the universe can never form at all, or does so, but collapses quickly. Of the stable solutions, they might have once existed, or yet might come into existence, or they might all co-exist in parallel with our own universe in what scientific speculators have called a multiverse. It means the universe is bigger than we thought because we only see our part of it, but much more is hidden from view, and this is called the multiverse to distinguish it from the older idea that the universe is what we see. The multiverse might be seething continuously with universes being spawned and dying like bubbles in a boiling pan of water. One of those bubbles is ours—the universe we see. This notion ties in with the idea in quantum theory of many virtual states which could appear according to the circumstances, but do not until something causes the wave equation to “collapse”, when the appropriate virtual state for the conditions will manifest itself. Virtual states seem to be possible universes.

The many possible states of the universe are all equally real. It explains the Anthropic Principle—that the laws and constants of our universe are so perfect for life. Physicists think even small changes in the value of constants would prevent life as we know it from ever forming. Atoms might not be able to form at all, or might be unstable and decay quickly, so starts would burn too fast, or perhaps not burn at all, or gravity would be too weak or too strong, or perhaps even negative making everything fly apart. The Anthropic Principle is simply that we live in a universe that is just right for us. Christians see the hand of God in it, but an almighty God ought not to be restricted at all, so such a constraint does not fit the idea of God being almighty. Quite literally, we would be like a fish out of water in any other universe. We could not live in it, and could not observe it. We see what we see because it was just right for us.

Countless billions of universes all unsuitable for life might have formed before ours was. Or, if we live in a multiverse in which other universes co-exist and spawn new universes, with slightly different physical constants, from black holes, then the universes could evolve until all of them were optimal. Short-lived ones would not produce enough universes like them, so that those with longer lives would predominate, spawning universes with longer lives. In an eternity, if that is how long the multiverse exists, all of the universes within it would tend towards the same type. Since ours must be one of the survivors, then all of them are like ours. All are able to make life and all can be observed when that life becomes intelligent. There is then nothing peculiar about our situation at all. It is the norm!

It [string theory] goes so far beyond our physical experience of the world that some critics say that it should be considered more as a work of philosophy than the ultimate scientific description of nature.

Alok Jha, The Guardian

A problem with string theory was that it came in five varieties, each of which seemed as good as any other, and it could not yield up space and time, as a proper TOE must. Space and time have to be assumed as being among the dimensions. Then in 1995, Ed Witten, a brilliant mathematics professor at Princeton, showed that the five varieties could be reconciled in a single hypothesis that encompassed all of them, and showed that they were just different views of the same basic theory—M-theory. String theory was a description in ten dimensions. M-theory simply added one extra one to give eleven, and by so doing extended the strings potentially into surfaces or membranes. An extension of string theory into eleven dimensions yilded a space containing two dimensional surfaces or membranes (branes) in the place of the one dimensional strings. The membranes still vibrate but now they are like the vibrations of the skin of a drum.

On the face of it, though, the extension is not attractive because it seems to extend the already huge number of possibilities. Then Witten took a further step by introducing twistor theory, an idea developed by another mathematical genius, Roger Penrose decades ago at Oxford University. It seemed to give similar results without the need for all those extra dimensions. It is, though, as yet imperfectly formulated. Work is still needed on it. Loop Quantum Gravity (LQG) is an approach which does yield space and time, and also shows that space is not continuous at the scale of the Planck length of 10^-35 meters. The problem with it is that it is restricted to the quantum scale of things and cannot bring in gravity. Quantum Information Theory and Qubit Field Theory are also approaches. The universe might be simply the effect of a large number of on/off switches, like Conways game of life. It predicts that, at smaller scales than the Planck length, space gets smooth again.

To distinguish these different approaches more data are needed, but that means building vast and expensive particle accelerators, like the Large Hadron Collider at CERN in Switzerland, or looking more closely at the galactic universe where all of these forces are being exerted. The theory suggests that particles should have supersymmetric equivalents at higher energies—an electron has a supersymmetric “selectron”, a quark a supersymmetric “squark”, and so on.

String theory might ultimately prove to be the TOE. In quantum theory and relativity, space and time are givens, but string theory has no need for such assumptions. One of the tests of its success will be that it yields space and time. The theory will yield the fundamental basis of our material world, and so it will be the TOE. The analogy of strings might break down, but whatever the fundamental things are, they will be truly fundamental.

All of this might seem like the empty musings of the medieval scholastics, but as long as suitable data can be collected by cosmological observations or from high energy machines, the speculations can be tested. So far as believers are concerned, the physicists are learning what is in the mind of God! That is what they do not like.

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